Download Particle Statistics Affects Quantum Decay and Fano Interference

PRL 114, 090201 (2015)
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PHYSICAL REVIEW LETTERS
Particle Statistics Affects Quantum Decay and Fano Interference
Andrea Crespi,1 Linda Sansoni,2,* Giuseppe Della Valle,3,1 Alessio Ciamei,2 Roberta Ramponi,1,3 Fabio Sciarrino,2,†
Paolo Mataloni,2 Stefano Longhi,3,1,‡ and Roberto Osellame1,3,§
1
Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
2
Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy
3
Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
(Received 6 October 2014; published 2 March 2015)
Quantum mechanical decay, Fano interference, and bound states with energy in the continuum are
ubiquitous phenomena in different areas of physics. Here we experimentally demonstrate that particle
statistics strongly affects quantum mechanical decay in a multiparticle system. By considering propagation
of two-photon states in engineered photonic lattices, we simulate quantum decay of two noninteracting
particles in a multilevel Fano-Anderson model. Remarkably, when the system sustains a bound state in the
continuum, fractional decay is observed for bosonic particles, but not for fermionic ones. Complete decay
in the fermionic case arises because of the Pauli exclusion principle, which forbids the bound state to be
occupied by the two fermions. Our experiment indicates that particle statistics can tune many-body
quantum decay from fractional to complete.
DOI: 10.1103/PhysRevLett.114.090201
PACS numbers: 42.82.Et, 03.65.Xp, 03.67.Ac, 42.50.−p
Decay of excited states has been a topic of great interest
since the early times of quantum mechanics [1]. A
metastable state decaying into a continuum usually shows
an exponential decay law and is modeled by a Breit-Wigner
resonance, which is the universal hallmark of unstable
states. However, more complex manifestations of quantum
mechanical decay are observed in the strong coupling
regime or when different decay channels are allowed.
Strong coupling is responsible for memory effects, leading
to deviations from an exponential decay law [2,3], whereas
multipath decay channels can lead to Fano interference and
asymmetric line shapes (Fano resonances [4,5]). Fano’s
model [4,5] is a landmark in modern physics. First
developed to explain the behavior of electrons scattered
by excited atoms [4], it was later adopted to explain
phenomena in a number of different physical systems
[6], such as ultracold gases and Bose-Einstein condensates
(where the Fano resonance is usually referred to as
Feshbach resonance [7]), semiconductors, quantum dots
and mesoscopic systems [8–10], and plasmonic nanostructures [11]. Fano interference is observed when different decay channels interfere, giving rise to broadening and
asymmetric deformations of natural line shapes. In general,
the destructive interference between different decay channels is associated with the formation of bound states in
the continuum [4,12], which inhibit the complete decay of
the excited state. The interplay between bound states in
the continuum and Fano-Feshbach resonances has been
highlighted in several works (see, for instance, [13–16]).
Quantum decay processes [2,3], Fano resonances [4,6],
and bound states in the continuum [17–21] have been
so far demonstrated in several systems using particles with
either bosonic (e.g. neutral atoms, photons) or fermionic
0031-9007=15=114(9)=090201(5)
(e.g. electrons) nature. However, in such previous experiments the peculiar role of particle statistics in the decay
process was not disclosed. Interestingly, recent works
[22–28] showed that particle statistics and contact interactions can deeply modify the decay dynamics in a manybody system. Even in the absence of particle interaction,
fermions and bosons may show very different decay
behavior, in particular in many cases fermions tend to
decay faster [22–24]. However, no experimental observation of this phenomenon has been reported yet.
In this work we investigate, experimentally, the decay
process of two noninteracting particles to a common
continuum, by probing an engineered photonic lattice
with two-photon states. The lattice, consisting of a threedimensional array of coupled optical waveguides, is
fabricated in a glass substrate by femtosecond laser
micromachining [29–31]. While the bosonic dynamics is
naturally observed for identically polarized photons, an
antisymmetric polarization-entangled state of the two
photons is used to simulate the fermionic behavior
[32–34], with anyonic statistics in the intermediate regime
[32,33]. Our experimental results indicate that particle
statistics by itself (considering noninteracting particles)
is sufficient to tune many-body quantum decay from
fractional to complete when the statistics is changed from
bosonic to fermionic, respectively.
We focus on systems described by the Fano-Anderson
[4,6] or Friedrichs-Lee Hamiltonian [35,36], which is a
paradigmatic model to study quantum mechanical decay,
Fano interference phenomena, and bound states in the
continuum [12,15,16,27]. To investigate the role of particle
statistics, second quantization of the Fano-Anderson
Hamiltonian is required [27]. The simplest case is provided
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© 2015 American Physical Society
PHYSICAL REVIEW LETTERS
PRL 114, 090201 (2015)
by two discrete states coupled to a common tight-binding
continuum of modes, i.e. a quantum wire, which are
initially occupied by two indistinguishable particles, either
bosons or identical fermions [27]. In detail, we consider
a system composed of two sites j1i and j2i, respectively,
with energy ϵ1 and ϵ2 , side coupled with hopping rates κ 1
and κ 2 , to a common semi-infinite chain of coupled sites
(a quantum wire), each with energy ϵ ¼ 0 [see Figs. 1(a)
and 1(b)]. Thus, states j1i and j2i can decay by tunnelling
to the common continuum given by the tight-binding lattice
band of the quantum wire. The energy of the band spans the
interval −2κ < E < 2κ, κ being the hopping rate between
two adjacent sites of the wire. In the following, ϵ1 and ϵ2
will be considered embedded into the continuum, i.e.
jϵ1;2 j < 2κ, and the coupling of the sites j1i and j2i to
the continuum will be assumed as weak, i.e. κ 1;2 < κ.
In our experiment, the sites correspond to optical modes
of different waveguides, which interact through their
evanescent field. The system is thus formed by two
waveguides coupled to a linear waveguide array, according
to the geometry of Fig. 1(c). The coupling coefficients κ, κ 1 ,
κ2 depend on the distance between the waveguides and
can be tailored by carefully dimensioning the structure.
The energies ϵ1 , ϵ2 , and ϵ correspond to the propagation
(a)
(b)
(c)
constants of the waveguides themselves and can be tuned
by varying the refractive index change in the different
waveguides (for our direct laser writing process, this is
achieved by tailoring the writing speed). In this photonic
implementation, the temporal dynamics of the system is
mapped onto the propagation distance z.
The evolution of the particle-creation operators a†j for the
various modes is governed by the coupled-mode equations:
i
da†1;2
¼ ϵ1;2 a†1;2 þ κ1;2 a†3 ;
dz
da†
i 3 ¼ κ1 a†1 þ κ 2 a†2 þ κa†4 ;
dz
da†j
i
j ≥ 4:
¼ κa†j−1 þ κa†jþ1 ;
dz
ð1Þ
where a†1 and a†2 refer to the modes j1i and j2i, while a†j
with j > 2 refer to the modes of the linear array. The
equivalence between the semi-infinite lattice model,
described by the operator equations (1), and the FanoAnderson model can be readily established by an operator
transformation from the Wannier to the Bloch basis
representation [15,16,27]. An interesting property of this
system is the existence of one bound state in the continuum
when ϵ1 ¼ ϵ2 . In fact, it is easy to observe that in this
case the operator b† ¼ eiϵ1 z ða†1 =κ 1 − a†2 =κ 2 Þ satisfies
db† =dz ¼ 0; i.e. the population of the dressed state
described by b† does not decay. In the Bloch basis of
operators, the bound state can be interpreted as a result
of a destructive Fano interference between different decay
channels [15,16,27]. This leads to fractional decay when a
single particle (photon) is placed in either site (waveguide)
j1i or j2i. We are studying here the two-particles case,
considering an initial state jΨð0Þi of the system excited
with one photon in j1i and one photon in j2i. Quantum
decay is described by the survival probability:
PS ðzÞ ¼ jhΨð0ÞjΨðzÞij2 ;
FIG. 1 (color online). Schematic of (a) two quantum wells side
coupled to a tight binding quantum wire, and (b) representation
of the energy levels. (c) 3D rendering of the actual photonic
structure employed in the experiments; two waveguides coupled
to a vertical linear array represent the two discrete states coupled
to the continuum. The two-photon state is launched in the
photonic structure as indicated by the red arrows.
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ð2Þ
which is the probability that, at a propagation distance z,
both particles are still in the initial state and none has
decayed into the continuum. From an experimental point
of view, PS corresponds to the probability of coincidence
detection of two photons in output modes j1i and j2i.
By exciting the system with two identically polarized
photons, the natural bosonic behavior is observed. On the
contrary, if the system is excited with an antisymmetric
polarization-entangled two-photon state, PS ðzÞ has the
same expression as if we were injecting two identical
particles with fermionic statistics; i.e. the operators a†j
satisfy fermionic commutation rules (see Refs. [32,33] and
the specific discussion in the Supplemental Material [37]).
Figures 2(a) and 2(b) show numerical simulations of
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(a)
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PHYSICAL REVIEW LETTERS
PRL 114, 090201 (2015)
(b)
previous states, a temporal delay for one of the photons).
These conditions correspond to identical bosons, identical
fermions, and distinguishable particles. The survival probability for identical bosons PS;bos and fermions PS;fer is then
retrieved from experimental measurements as follows:
PS;bos ¼
FIG. 2 (color online). Numerical simulations showing the
behavior of the survival probability PS for different propagation
lengths and energy detuning, normalized with respect to κ, for
the case of (a) bosonic and (b) fermionic particles. Parameter
values are κ1 ¼ κ 2 ¼ 0.37κ and ϵ1 ¼ 0.926κ, which apply to our
experiment.
survival probability PS ðzÞ for a system described by (1), in
normalized coordinates, for initial two-particle states either
bosonic or fermionic. A striking difference between the
bosonic and fermionic behavior is evident, as will be
experimentally demonstrated in the following.
As a first experiment, we concentrate on the investigation of the quantum decay in the case ϵ1 ¼ ϵ2 . We
have fabricated several structures as that reported in
Fig. 1(c), yielding κ1 ¼ κ 2 ¼ 0.2 mm−1 , κ ¼ 0.54 mm−1 ,
ϵ1 ¼ ϵ2 ¼ 0.5 mm−1 , and different lengths z of the array.
In our realization the linear array is composed of 25
waveguides. Details on the fabrication process by femtosecond laser direct writing are given in the Supplemental
Material [37]. Each structure allows us to photograph the
evolution at a specific propagation distance z.
The system is first characterized by launching laser light
separately in waveguides j1i and j2i, and imaging the
output facet with a digital camera. The fraction of light
remaining in the launch waveguides is measured. This
corresponds to investigating the single-particle behavior,
when the particle is initially on mode j1i or j2i. Survival
probability PS;clas for two classical, distinguishable particles is easily calculated from the product of two singleparticle experimental distributions (since the two particles
are uncorrelated).
To experimentally characterize the system behavior for
two correlated particles, two photons at 810 nm wavelength, generated by a spontaneous parametric downconversion source, are coupled to single-mode optical
fibers and injected simultaneously in waveguides j1i and
j2i. Output light from the same waveguides is collected by
an objective, coupled to multimode fibers and detected
by single-photon avalanche photodiodes [see Fig. 3(a)].
Coincidence-detection counts, in equal temporal gates, are
performed for different input states: indistinguishable
vertically polarized photons, polarization-entangled photons in an antisymmetric state, and distinguishable photons
(the latter being generated by introducing, for each of the
CVV
P
;
CVV;dist S;clas
PS;fer ¼
Cent
CVH;dist
PS;clas ; ð3Þ
where CVV are the coincidence counts for the jVVi state,
CVV;dist the corresponding counts when one photon is
delayed, Cent arepthe
ffiffiffi coincidence counts for the entangled
ðjHVi − jVHiÞ= 2 state, and CVH;dist the corresponding
counts when one photon is delayed (H and V refer to
horizontal and vertical polarization, respectively).
Figure 3(b) reports the experimentally characterized
survival probabilities for the different z and different input
states, compared to numerical simulations for the same
system. While PS;bos shows a fractional decay owing to the
existence of a bound state, PS;fer shows a full decay.
The experimental points for fermions do not reach exactly
zero due to imperfections in preparing the entangled state
and slight polarization dependence of the photonic device.
Imperfections in the entangled state preparation could
be both in the π phase, giving the minus sign in the
antisymmetric state, and in a partial photon distinguishability. Respectively, these features induce a partial bosonic
(precisely anyonic [32]) or partial classical behavior, both
of which introduce a nonvanishing survival probability
of the initial state. However, such results clearly show that
fractional decay is suppressed for two fermions. This is a
signature of the Pauli exclusion principle and can be
explained by observing that no more than one fermion
can be accommodated into the dressed bound state, while
the other fermion necessarily decays into the state
continuum.
To better highlight the difference between bosonic and
fermionic behavior in the decay process, Fano-like profiles
are measured from the survival probability as the detuning
of the energy levels of the two discrete states is varied
[15,16] (see the Supplemental Material [37] for more
technical details). We fabricated and characterized other
photonic structures, with fixed length z ¼ 20 mm, the same
κ1 , κ 2 , κ as in the previous experiments, ϵ1 ¼ 0.5 mm−1 and
different values for ϵ2. Results are shown in Fig. 3(c).
For bosonic particles, the survival probability shows a
resonance behavior peaked at ϵ1 ¼ ϵ2 . This peak is
associated with fractional decay observed in the previous
experiment and corresponds to perfect destructive interference of the decay channels into the common continuum.
At ϵ2 − ϵ1 ≃ 0.255 mm−1 , the survival probability shows
a minimum, corresponding to a maximally constructive
interference of the decay channels and the largest
decay rate (see the Supplemental Material [37]). A similar
behavior is found for distinguishable particles. Interestingly
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PHYSICAL REVIEW LETTERS
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(a)
(b)
(c)
FIG. 3 (color online). (a) Experimental setup for two-photon measurements: two-photon entangled states are generated by
spontaneous-parametric down-conversion in a beta-barium borate (BBO) crystal. The generated state is tuned by wave plates
(WP) and a liquid crystal retarder (LC); photons are coupled to single-mode fibers (SMF) and injected into the chip. A microscope
5 × objective (OBJ) collects the output light and images the two lateral modes directly onto the entrance facets of two multimode fibers
(MMF) in a 250 μm pitched fiber array. The fibers are connected to single-photon avalanche photodiodes (SPAD) for detection. A delay
line (DL) controls the temporal indistinguishability of the photons. Measurements for jVVi state are performed by injecting a
p1ffiffi ðjHHi þ jVViÞ state in the chip and postselecting with a polarizer (POL). (b),(c) Experimental maps of the survival probability for
2
two bosons (red circles), two fermions (blue diamonds), two distinguishable particles (black crosses). Curves of numerical simulation
are also shown. In (b) ϵ2 ¼ ϵ1 ¼ 0.5 mm−1 and z is varied, which corresponds to the vertical dashed lines in the maps of Fig. 2. In
(c) z ¼ 20 mm is kept fixed while ϵ2 is varied, which corresponds to the horizontal dashed lines in the maps of Fig. 2. Where not shown,
error bars are smaller than marker size. The two arrows in (c) correspond to the energy detunings of destructive ðϵ2 ¼ ϵ1 Þ and maximally
constructive interference (ϵ2 − ϵ1 ≃ 0.255 mm−1 ) of the decay channels.
the resonance peak is absent when the two particles possess
fermionic statistics, since in this case Pauli exclusion
principle forbids the two fermions to be accommodated
in the bound state. Such a result is a rather general one and
holds whenever the number of bound states in the continuum of the Fano-Anderson Hamiltonian is lower than the
number of particles [27].
In conclusion, this work has presented an experimental
study on the quantum decay process of two identical
particles, initially on discrete states, into a common
continuum. A profound difference in the bosonic and
fermionic evolution is evidenced; in particular, we found
that quantum decay can be tuned from fractional to
complete by changing the particle statistics from bosonic
to fermionic. This system can be seen as a particle statistics
filter, in fact, considering the two initial particles of any
statistics, either bosonic or fermionic, after a sufficiently
long time only the particles of bosonic statistics will survive
in the initial state. The capability to simulate multiparticle
dynamics in discrete systems coupled to a continuum
may enable the investigation of other multiparticle decay
phenomena, such as multiparticle Zeno effects [39] and
non-Markovianity [40]. In addition, the current approach
is not only limited to two particles; in fact, proper
multiphoton entangled states could be used to reproduce
the dynamics of an arbitrary number of bosons or
fermions [33].
This work was supported by ERC (European Research
Council) Starting Grant 3D-QUEST (3D-Quantum
Integrated Optical Simulation, Grant Agreement
No. 307783; www.3dquest.eu) and by the European
Union through the project FP7-ICT-2011-9-600838
(QWAD Quantum Waveguides Application and
Development; www.qwad-project.eu). Partial support from
the Fondazione Cariplo is also acknowledged by G. D. V.
and S. L. through the project New Frontiers in Plasmonic
Nanosensing (Grant No. 2011-0338).
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*
Current affiliation: Integrated Quantum Optics, University
of Paderborn, Warburger Strasse 100, 33098 Paderborn,
Germany
†
[email protected]
PRL 114, 090201 (2015)
‡
[email protected]
[email protected]
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